On Dec 9, 12:24 pm, Zuhair <zaljo...@gmail.com> wrote:> On Dec 9, 10:59 pm, WM <mueck...@rz.fh-augsburg.de> wrote:>>>> > The real numbers as we teach them are at issue. Everything else may be> > left to the "experts" or fools of matheology.>> > Regards, WM>> The real numbers you teach can be presented by countable models, and> also can be proved uncountable in other models.>> Zuhair

Measure theory, that we use for all standard results, has yetcountable additivity. The partition of the segment is to countablymany partitions, adding those back up gives us the integral, the areaunder the curve.

That there are even non-measurable sets of reals, is from Vitali'sargument that there would be some infinitesimal constant, Vitali's c,the sum of which over the naturals is two (or between one and three).

Measure theory doesn't need re-Vitali-ization to see that the resultsare courtesy: countable additivity. As well Banach and Tarski's ball-doubling might see much more realm for application, given onlyslightly different first principles.

The paths of the tree are of the nodes, with the rationals being quitelarge.

Yes, the structure of transfinite cardinals is a mathematicalabstraction tractable to our devices of reason, but, nobody'sdiscovered applications for them yet, for real analysis (or physics).And: dependence on them as the foundation: closes the door on anyconsideration of considering the points: falling in a line, in theirnatural order, for what they do.

Simply enough, Cantor's results are true in that the line can't bedrawn, in the graphical and the intuitive sense and as a plainprojection of the space, without drawing them (its points) in order.And, they're important in allowing to mathematics the infinite andtransfinite ordinals, and relevant transfinite induction, for theordinals besides the cardinals. And, they do establish a relevantordering of infinite digital sets, but not the only one nor for thatmatter one unavailable to the construction in the ordinals, simply amore direct one. (Half of the integers are even.) And, they'reimportant as a part of the historical development, yet another chapterin the discussion since antiquity, of the realm of thought.

The Universe as it exists would be its own powerset, and Cantor didsee an Absolut infinity in his Mengenlehre, and ZFC is post-Cantorianwith infinite ordinals. Yggdrasil: there's an eagle on top, and theeagle on top of it. There's nothing to contain the universe butitself (re Kant's the Ding-an-Sich). And Cantor, Georg, also wanted acompleted infinity he could count from, toward the origin, not justto, from the origin, in his own words. And: it's turtles.

Most all our stories start with nothing and go from there. Thatanswers a real deep question.